Three (or more difficult version: any number $n>2$) points move on a unit circle with pairwise different constant speeds. For which initial points and which speeds is the following true:
for every $\epsilon > 0$ there is a point in time at which all the $3$ (or $n$) points are contained in a ball with radius $\epsilon$
(then they will obviously move apart again, so I'm not talking about convergence, but rather convergence up to a subsequence in time)?
It's pretty easy to prove that for $n=3$ as long as the speeds $a,b,c$ satisfy $(a-b)/(c-b) \notin \mathbb{Q}$, then the answer is affirmative. It's also pretty easy to see that if any two subsets of points with positions $x_1, ..., x_n, y_1, ..., y_m$ and the quotien of their sums of speeds $X/Y$ satisfy $X/Y = n/m$ and $m\cdot\sum x_i - n\cdot\sum y_i \neq 0 $, then the answer is negative (since with the assumtion on the quotient of sums of speeds the last difference doesn't change).